1. Singular Hodge theory of matroids; 2. Logarithmic concavity of weight multiplicities for irreducible sln(C)-representations

Talk 1: Title: Singular Hodge theory of matroids

If you take a collection of planes in R^3, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the “Top-Heavy Conjecture”, that Dowling and Wilson conjectured in 1974.

On the other hand, given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan–Lusztig polynomial) to it.

The problems of proving the “Top-Heavy Conjecture” and the non-negativity of the coefficients of these Kazhdan–Lusztig polynomials are related, and they are controlled by the Hodge theory of a certain singular projective variety. The “Top-Heavy Conjecture” was proven for hyperplane arrangements by Huh and Wang in 2017, and the non-negativity was proven by Elias, Proudfoot, and Wakefield in 2016. I will discuss work, joint with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang, on these two problems for arbitrary
matroids.

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Talk 2: Logarithmic concavity of weight multiplicities for irreducible sln(C)-representations

Log-concavity is a combinatorial property that can sometimes hint toward an underlying Hodge-theoretic structure. We will show that the sequence of weight multiplicities we encounter is log-concave, as we walk along any root direction in the weight diagram of a finite-dimensional irreducible representation of sln(C).  Along the way, we prove both a continuous and discrete kind of log-concavity for Schur polynomials, and we conjecture that many related polynomials in algebraic combinatorics also exhibit log-concavity phenomena. As a consequence of our results, we obtain a special case of Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients. This is joint work with June Huh, Karola Meszaros, and Avery St. Dizier.